Simplified Inverse Method for Determining the Tensile ...

Journal of Advanced Concrete Technology Vol. 5, No. 2, 235-246, June 2007 / Copyright © 2007 Japan Concrete Institute 235

Scientific paper

Simplified Inverse Method for Determining the Tensile Strain Capacity of Strain Hardening Cementitious Composites Shunzhi Qian1 and Victor C. Li2

Received 11 December 2006, accepted 27 April 2007

Abstract As emerging advanced construction materials, strain hardening cementitious composites (SHCCs) have seen increasing field applications recently to take advantage of its unique tensile strain hardening behavior, yet existing uniaxial tensile tests are relatively complicated and sometime difficult to implement, particularly for quality control purpose in field applications. This paper presents a new simple inverse method for quality control of tensile strain capacity by conduct-ing beam bending test. It is shown through a theoretical model that the beam deflection from a flexural test can be line-arly related to tensile strain capacity. A master curve relating this easily measured structural element property to mate-rial tensile strain capacity is constructed from parametric studies of a wide range of material tensile and compressive properties. This proposed method (UM method) has been validated with uniaxial tensile test results with reasonable agreement. In addition, this proposed method is also compared with the Japan Concrete Institute (JCI) method. Compa-rable accuracy is found, yet the present method is characterized with much simpler experiment setup requirement and data interpretation procedure. Therefore, it is expected that this proposed method can greatly simplify the quality con-trol of SHCCs both in execution and interpretation phases, contributing to the wider acceptance of this type of new ma-terial in field applications.

1. Introduction

In the past decade, great strides have been made in de-veloping strain hardening cementitious composite (SHCC), characterized by its unique macroscopic pseudo strain hardening behavior after first cracking when it is loaded under uniaxial tension. SHCCs, also referred to as high performance fiber reinforced cemen-titious composites (HPFRCCs, Naaman and Reinhardt 1996), develop multiple cracks under tensile load in contrast to single crack and tension softening behavior of concrete and conventional fiber reinforced concrete. Multiple cracking provides a means of energy dissipa-tion at the material level and prevent catastrophic frac-ture failure at the structural level, thus contributing to structural safety. Meanwhile, material tensile strain hardening (ductility) has been gradually recognized as having a close connection with structural durability (Li 2004) by suppressing localized cracks with large width. Many deterioration and premature failure of infrastruc-ture can be traced back to the brittle nature of concrete. Therefore, SHCCs are considered as a promising mate-rial solution to the global infrastructure deterioration problem and tensile ductility is the most important prop-erty of this type of material.

Engineered Cementitious Composites (ECC, Li 1993) is a unique representative of SHCCs, featuring superior ductility (typically > 3%, 300 times that of normal con-crete or FRC) (Li and Kanda 1998; Li et al 2001), tight crack width (less than 80µm, Li 2003), and relatively low fiber content (2% or less of short randomly oriented fibers). A typical tensile stress-strain curve of ECC is shown in Fig. 1. It attains high ductility with relatively low fiber content via systematic tailoring of the fiber, matrix and interface properties, guided by microme-chanics principles. Enhanced with such high tensile duc-tility and/or tight crack width, ECC has demonstrated superior energy dissipation capacity, high damage toler-ance, large deformation capacity, and exceptional dura-bility in many recent experimental investigations (Li 2005). As a result, ECC is now emerging in the field and has seen increasing infrastructure applications, such as dam repair, bridge deck overlay and link slab, coupling beam in high-rise building, and other structural elements and systems (Li 2004).

As aforementioned, tensile ductility is the most im-portant material property of SHCC, yet relatively large variation of tensile ductility was observed in the litera-ture (Kanda et al 2002, 2006; Wang and Li 2004). To address such concern, Wang and Li (2004) have pro-posed using artificial flaws with prescribed size distri-bution as defect site initiator to create more saturated multiple cracks, resulting in more consistent tensile strain capacity among different specimens from the same batch. The overall tensile strain capacity shows much more consistent results after implantation of arti-ficial flaws, however, the variation of tensile strain ca-pacity is still relatively large when compared with that

1Research Assistant, Department of Civil and Environmental engineering, University of Michigan, USA. 2Professor, Department of Civil and Environmental engineering, University of Michigan, USA. E-mail: [email protected]

236 S. Qian and V. C. Li / Journal of Advanced Concrete Technology Vol. 5, No. 2, 235-246, 2007

of other properties, e.g., first cracking strength. There-fore, test method for quality control of SHCCs onsite should logically focus on tensile strain capacity due to its importance in governing structural response and po-tentially large variability.

While most characterization of the tensile behavior of SHCCs was carried out using uniaxial tensile test (UTT) in academia, this method is generally considered to be complicated, time-consuming and require advanced equipment and delicate experimental skills. Therefore, it is not suitable for onsite quality control purpose (Stang and Li 2004, Ostergaard et al 2005, Kanakubo 2006). First, special fixtures and/or treatments for the ends of specimens are usually needed in order to transfer tensile loads. Furthermore, the specimen is sensitive to stress concentration induced by misalignment and can fail near the end prematurely. Last but not least, realistic dimensions for specimens large enough to have 3-dimensional random fiber orientation make the UTT even more difficult to conduct.

As a simpler alternative to the UTT, four point bend-ing test (FPBT) was proposed by Stang and Li (2004) for quality control on construction sites, provided that an appropriate interpretation procedure for the test result is available. FPBT, in which the mid-span of the speci-men undergoes constant bending moment, may be car-ried out to determine the moment-curvature or moment-deflection curves. This type of test is much easier to set up and conduct in comparison to UTT, and a large amount of experience in bending test has been accumu-lated in the user community of cementitious materials. The ultimate goal of this test is to use the moment-curvature or moment-deflection curves so determined to invert for the uniaxial tensile properties. It should be noted, however, that the bending test is not meant to determine whether the material has tensile strain-hardening behavior or tension-softening behavior, but rather to constrain the tensile material parameters, e.g. the tensile strain capacity, as part of the quality control process in the field.

Inverse analyses for FPBT have recently been at-tempted by Technical University of Denmark (DTU)

and Japan Concrete Institute (Ostergaard et al 2005; Kanakubo 2006) with certain success. By adopting a simplified elastic-perfectly plastic tensile model, JCI method generally can predict plateau tensile strength and tensile strain capacity from the FPBT results via a sectional analysis similar to that developed by Maalej and Li (1994). On the other hand, hinge model, includ-ing both tensile strain hardening and tension softening effect, was employed in the DTU inverse method along with least square method to invert for tensile material properties from their bending response. The model can predict experimental load – deflection curve fairly well and tensile properties derived based on this method agree well with that from FEM analysis, yet no direct comparison with UTT results has been made so far.

Despite the successes mentioned above, further sim-plification and/or validation are necessary to make the FPBT widely accepted for quality control of SHCCs. In case of JCI method, significant improvement is needed to simplify the experimental execution and data inter-pretation procedure. For instance, LVDTs are required in JCI method to measure the beam curvature. This is somewhat burdensome in field conditions, considering quality control may involve a large number of speci-mens. Furthermore, the inverse process is not user friendly, which require relatively complicated calcula-tion (solving cubic equation). As for the DTU method, firstly it needs complementary UTT results to truly vali-date the model. Secondly, the uniqueness of solution from such inverse analysis is questionable at times. Fi-nally, the method will need to be packaged into sophis-ticated software, which may incur additional user cost. A simple engineering chart with reasonable accuracy may be more preferable.

Keeping these considerations in mind, this paper looks to develop a greatly simplified yet reasonably accurate inverse method for determining tensile strain capacity of SHCCs. In the following sections, the re-search significance and overall research framework for the proposed method will be presented first, followed by parametric study to obtain the master curves for inverse analysis. Thereafter, the experimental program consist-ing of both FPBT and complementary UTT will be re-vealed in detail. The results from FPBT will then be converted to tensile strain capacity and validated with independent UTT test results. Finally, the proposed method will be compared with JCI inverse method, fol-lowed by overall conclusions.

2. Overall research framework

The overall research framework is revealed in Fig. 2, consisting of detailed procedure for development and execution of inverse method (top frame (a)) and valida-tion and verification procedure (bottom frame (b)). As shown in the top frame (a), deflection capacity (deflec-tion corresponding to peak bending stress, i.e., modulus of rupture) can be obtained from FPBT. By conducting

0

1

2

3

4

5

6

7

0 1 2 3 4 5Tensile strain (%)

Tens

ile st

ress

(MPa

) .

Fig. 1 Typical tensile stress-strain curve of ECC.

S. Qian and V. C. Li / Journal of Advanced Concrete Technology Vol. 5, No. 2, 235-246, 2007 237

parametric studies based on a flexural behavior model of SHCCs, master curve can be constructed in terms of tensile strain capacity with respect to deflection capacity. Based on deflection capacity of FPBT and master curve from parametric study, tensile strain capacity of SHCCs can be derived. Additionally, companion UTT test using specimens cast from the same batch of material is used to validate and/or verify the proposed method in terms of the accuracy of derived tensile strain capacity, which is shown in bottom frame (b). It is conceived that a large number of FPBT will be conducted at construction sites on daily basis to ensure the quality of the SHCC mate-rial. The extra step of verification (Frame (b)) will not be necessary once the method is standardized and util-ized in practice, except a very limited number of UTT specimens to be cast onsite and tested at advanced re-search laboratory to check whether this material is truly a strain hardening type during trial mixing stage. 3. Parametric study and master curves

3.1 Flexural behavior model The flexural behavior model used in this investigation is based on the work of Maalej and Li (1994). Compared with other models, the major distinction of this model is that the contribution of tensile strain hardening property of SHCCs was included. The actual SHCC considered in the model is Polyethylene ECC (PE-ECC) material. To simplify the analysis, the stress – strain behavior of the ECC was assumed as bilinear curves in both tension and compressive. Based on a linear strain profile and equilibrium of forces and moment in a section, the rela-tion between flexural stress and tensile strain at the ex-treme tension fiber (Simplified as critical tensile strain hereafter) can be determined as a function of basic ma-terial properties. Overall, the model predicts experimen-tally measured flexural response quite well. For more detail, the readers are referred to Maalej and Li (1994).

Based on geometrical considerations, the beam curva-

ture can be computed as the ratio of critical tensile strain to the distance from the extreme tension fiber to the neutral axis. This can be expressed in following equa-tion:

ctε

ρφ ==

1 (1)

whereφ , ρ , tε , and c are beam curvature, beam radius of curvature, critical tensile strain, and the distance from the extreme tension fiber to the neutral axis.

In a FPBT of SHCC material, according to structural mechanics, we can obtain following equation to relate the deflection of the beam to the curvature at the load point and therefore critical tensile strain. The load point

(b)

(a)

FPBT Derived tensile strain

capacity

Flexural stress – load point deflection curve

Tensile strain capacity – deflection capacity relation (master curve)

Flexural model (Maalej & Li 1994)

UTT Experimental tensile stress – strain relation

Validation/Verification

L

u u h

Fig. 3 Definition of load point displacement u, span length L, beam height h and curvatureφ in a schematic FPBT test (curvature is constant along the middle third span between the two load points).

φρ 1=

Fig. 2 Overall research framework for proposed UM method.


deflection for a beam with span length L is given by:

222

)(4*

162**15

LhLLu φφ

+= (2)

The definitions for u,φ , L and h are shown in Fig. 3. The first term includes the flexural deformation from pure bending span and shear span while the second term includes the shear deformation from shear span. It should be noted that the shear deformation is strongly related with the ratio of beam height over span length as shown in the second term. The specimens used in this study have a height and span length of 51mm, 305mm while corresponding values are 100mm and 300mm for JCI method. Therefore, the second term in Equation (2) for the UM method is at least an order of magnitude smaller than the first term, so that (2) can be simplified as follows:

2**1.0 Lu φ= (3)

Since the relation between flexural stress and tε is already established, we can predict the flexural stress and load point deflection relation based on Equations (1) and (3). The maximum flexural stress (MOR) and corresponding deflection (deflection capacity) are reached once the strain capacity of the SHCCs is ex-hausted either at the extreme tensile fiber or at the ex-treme compression fiber, which is the assumed failure criterion in this model. The strain capacity is defined as tensile/compressive strain corresponding to ultimate tensile/compressive strength (maximum stress in stress-strain curve).

The model was originally developed based on mate-

rial behavior of PE-ECC, it would be desirable to check its applicability for other SHCCs, such as PVA-ECC, which has been extensively investigated in the past dec-ade. PVA-ECC 0, with mix proportion and material ten-sile and compressive properties shown in Table 1, was utilized to check the applicability of the model. As shown in Fig. 4, the analytical result from the model predicts the experimental flexural stress-deflection curves reasonably well, particularly in terms of average MOR and deflection capacity prediction. This seems to suggest that the model is quite versatile in predicting the flexural behavior of SHCCs.

3.2 Construction of master curves Parametric study was conducted to investigate the influ-ence of material uniaxial tensile and compressive prop-erties (parametric values) on the flexural response of SHCCs based on the aforementioned flexural model. The correlation between tensile strain capacity and load point deflection was established and constructed as mas-ter curve. All tensile and compressive properties were varied within a wide range of parametric values (Table 2), covering the normal range of test results of SHCC specimens at UM and JCI (Kanakubo 2006). It is ex-pected that the master curves based on this wide range of parametric study can be directly utilized for quality control purpose in field.

With material parameters shown in Table 3, five cases of parametric study were plotted in Fig. 5 as ex-amples. The material parameters varied in first cracking strength, ultimate tensile strength, modulus of elasticity and compressive strength compared with Case 1 in Ta-ble 3. Figure 5 shows the flexural stress, load point

Table 1 Mix proportion and material properties of PVA-ECC 0.

Mix proportion Material properties

Cement Sand Fly Ash Water Super-

plasticizer Fiber

First cracking strength(MPa)

Ultimatetensile

strength(MPa)

Tensile strain

capacity (%)

Compressive strength (MPa)

1 1 0 0.44 0.007 0.02 5.1±0.2 5.7±0.5 2.0±1.0 50.7±5.3 Note: Mix proportion is by weight except PVA fiber by volume. The modulus of elasticity and the ultimate com-pressive strain are assumed to be 18 GPa and 0.005, respectively.

Table 2 Range of material parameters used in parametric studies to construct the tensile strain capacity – deflection capacity (curvature) relation.

Tensile properties Compressive properties

Material parameters

First cracking strength (MPa)

Ultimate tensile

strength (MPa)

Tensile strain

capacity(%)

Modulus of

elasticity (GPa)


Compressive strain capacity

(%)

Range 2.5~13.0 4.0~16.0 0~5 12~53 31~200 0.5~1*

Note: Parameters are in the normal range of test results of SHCC specimens at UM and JCI; Tensile and compres-sive modulus of elasticity are assumed to be equal; Beam dimensions are 51x76x356mm/100x100x400mm with span length of 305mm/300mm for UM and JCI specimens, respectively; *: Estimated range.


deflection and corresponding critical tensile strain rela-tion. Beam dimensions are 51x76x356mm with span length of 305mm. From the Figure, load point deflec-tions were observed to correlate very well with critical

tensile strains, regardless of the actual parametric mate-rial properties. Once the critical tensile strain reaches the tensile strain capacity, the beam reaches peak load and the corresponding load point deflection is the de-flection capacity. Therefore, it appears that the deflec-tion capacity and tensile strain capacity can be linearly correlated regardless of the values of other material properties.

The overall results from the parametric study indeed show a linear relation between tensile strain capacity and deflection capacity, as revealed in Fig. 6 (a). Totally 20 cases were investigated in the parametric study, with the range of material parameters shown in Table 2. All linear curves lie in a narrow band regardless of the val-ues of other material properties, which suggests that the beam deflection capacity is most sensitive to tensile strain capacity for a fixed geometry. For ease of quality control on site, a master curve was constructed as a line with uniform thickness to cover all parametric case studies, as shown in Fig. 6 (b). The top edge of the mas-ter curve is made to coincide with the upper boundary of all curves in Fig. 6(a) for conservativeness.

Additionally, another master curve correlating tensile strain capacity with curvature was constructed by para-metric study in order to compare the proposed UM method with JCI method, in which ultimate bending moment and curvature was utilized to derive tensile strain capacity. The range of parametric values is the same as the aforementioned parametric study, as shown in Table 2. The dimension of specimen used in this pa-rametric study is 100x100x400mm, with a span length of 300mm (JCI-S-003-2005).

As expected, this set of master curve also character-izes a linear relation within a very narrow band regard-less of actual material properties (Figs. 7 (a) and (b)). Since curvature may be linearly correlated with deflec-tion using Equation (2), this master curve can be easily transformed into tensile strain capacity to deflection capacity relation, even though the slope should be dif-ferent from Fig. 6 due to different dimensions used in the two parametric studies. In the case when specimens with different dimensions have to be used for quality control, e.g. due to different fiber length, a different set of master curve should be constructed.

Table 3 Assumed material properties for different cases of SHCCs.

Tensile properties Compressive properties


Ultimatetensile

strength (MPa)

Tensile strain

capacity (%)

Modulus of

elasticity (GPa)


Compressive strain capacity

(%)

Case 1 4.0 5.6 5% 18 50 0.005 Case 2 4.0 5.6 5% 20 50 0.005 Case 3 4.0 5.6 5% 18 75 0.005 Case 4 5.0 6.6 5% 18 50 0.005 Case 5 4.0 4.0 5% 18 50 0.005

0

4

8

12

16

20

0 2 4 6 8Load point deflection (mm)

Flex

ural

stre

ss (M

Pa) .

Fig. 4 Comparison of experimental results and analyti-cal prediction of flexural stress-load point deflection for PVA-ECC 0.

0

4

8

12

16

20

0 2 4 6 8 10 12 14 16Load point deflection (mm)

Flex

ural

stre

ss (M

Pa) .

Fig. 5 Parametric study for SHCCs with different mate-rial parameters (Dashed line boxes include markers corresponding to same critical tensile strain (tensile strain at the extreme tension fiber); markers are plotted on 1% strain interval after 1% strain for all cases) (No-tice Case 1 and Case 2 are almost coincided).

Case 4 Case 3 Case 1 Case 2 Case 5

1% 2% 3% 4% 5% Critical tensile strain

Analytical prediction

Experimental results


3.3 Discussion on the unique behavior of mas-ter curves The master curve reveals a linear behavior within a nar-row band width. In the following, discussions will be presented to provide some understanding of this unique phenomenon.

As shown in Equations (1) and (2), for a given span length L, it can be observed that middle span deflection (u) and curvature (φ ) would be linearly related to tε if c (the distance from extreme tension fiber to neutral axis) is constant with respect to tε . It turns out that this is approximately correct when tε is larger than 1%, as shown in Fig. 8, where parametric results are plotted for five cases of SHCCs (Table 4) in terms of distance from extreme tension fiber to neutral axis with tensile strain relation. It should be noted that the vertical axis (dis-tance from extreme tension fiber to neutral axis) is nor-malized by the beam height.

On the other hand, it is observed that these normal-ized distance (normalized c value) near plateau are very close (ranging from 0.86 to 0.91) for cases with drasti-cally different material properties. This explains the

narrow band width of the master curve. The close pla-teau values for different cases may be due to the fact that most of the SHCC materials have a very high ratio between compressive strength and tensile strength, thus pushing the neutral axis very close to the extreme com-pressive fiber. Comparison between Fig. 8 and Table 4 suggests that the higher the ratio, the higher the normal-ized c value. Nevertheless, the small difference between these normalized c values suggests that narrow band width may characterize the master curve for most SHCC materials, in which ratio of compressive strength over tensile strength is about 10.

3.4 Use method of master curves Based on the master curves obtained from parametric study, the deflection capacity from simple beam bending test can be easily converted to material tensile strain capacity, with detailed procedure schematically shown in Fig. 6 (b) and explained below. Assuming a deflec-tion capacity of 8 mm with standard deviation of 0.5 mm already known from FPBT, the predicted average tensile strain capacity can be derived by taking the aver-aged strain value corresponding to 8 mm deflection. The

0

1

2

3

4

5

6

0 100 200 300 400 500 600 700 800Curvature (µ/mm)

Stra

in c

apac

ity (%

) .

(a)

0

1

2

3

4

5

6

0 100 200 300 400 500 600 700 800Curvature (µ/mm)

Stra

in c

apac

ity (%

) .

(b)

Fig. 7 (a) Tensile strain capacity –curvature capacity rela-tion obtained from parametric study (20 cases); (b) Mas-ter curve transforming curvature capacity into tensile strain capacity.

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16Deflection capacity (mm)

Stra

in c

apac

ity (%

) .

(a)

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16Deflection capacity (mm)

Stra

in c

apac

ity (%

) .

(b)

Fig. 6 (a) Tensile strain capacity –deflection capacity relation obtained from parametric study (20 cases); (b) Master curve transforming deflection capacity into tensile strain capacity.

Upper bound

Lower bound Predicted Deviation

Master curve

7.5 8.5

Average value


upper and lower bound of the predicted tensile strain capacity can be obtained by taking maximum or mini-mum strain value corresponding to 8.5 mm or 7.5 mm deflection, respectively. The predicted deviation of ten-sile strain capacity is simply the difference between the upper/lower bound and average value.

A set of equations has been developed to simplify the conversion procedure, as shown below, where Equations (4) and (5) can be used to calculate the average tensile strain capacity and its deviation, respectively.

22.050.0' −⋅= utu δε (4)

18.050.0 +⋅= SDPD (5)

where 'tuε is the predicted average tensile strain capacity

(%); uδ is the average deflection capacity obtained from FPBT (mm) (In Fig. 6 (b), uδ is assumed to be 8mm); PD is the predicted deviation for tensile strain capacity (%) considering the standard deviation of the deflection capacity, as shown in Fig. 6(b) and SD is the standard deviation of the deflection capacity (mm) (as-sumed to be 0.5 mm). To be conservative, the lower bound equals to the lowest strain capacity value corre-

sponding to SDu −δ , which is assumed to be 7.5mm. Likewise, the upper bound equals to the highest strain capacity value corresponding to SDu +δ (assumed to be 8.5mm). Therefore, the Predicted Deviation is the dif-ference of upper bound/lower bound with predicted av-erage tensile strain capacity.

It should be noted that this equation can only be ap-plied to specimen with the same geometry and same loading conditions as that used by the authors (see Ex-perimental Program section). Should any of these ge-ometry and/or loading conditions change, another set of master curves and corresponding conversion equations should be developed for that purpose. Once the pro-posed method (or its modified version) is standardized and widely accepted, there should be no need for change in geometry and loading conditions.

A similar procedure can also be used to convert the curvature to strain capacity for the specimens tested according to the JCI method. A set of equation has also been developed according to Fig. 7 to simplify the con-version procedure. Equations (6) and (7) can be used to calculate the average tensile strain capacity and its de-viation, respectively.

24.00093.0 ,, −⋅= cuctu φε (6)

19.00093.0 +⋅= cc SDPD (7)

where ctu ,ε is the predicted average tensile strain capac-ity (%); cu ,φ is the average curvature capacity obtained from FPBT (µ/mm); PDc is the predicted deviation for tensile strain capacity (%) and SDc is the standard devia-tion of the curvature capacity (µ/mm). The same limita-tion as mentioned above for Equations (4) and (5) also applies to Equations (6) and (7), except that the speci-men geometry and loading profile should follow those in the JCI method. 4. Experimental program

4.1 Materials, Specimen Preparation and Test-ing The mix proportion of SHCC materials investigated in

Table 4 Comparison of ratio of compressive strength over average tensile strength for different SHCCs (material properties are assumed parametric values).

Modulus of Elasticity

(GPa)


Ultimate tensile

strength (MPa)

Average tensile

strength (MPa)

Comp. strength (MPa)

Ratio of comp. strength/Average tensile strength

SHCC 1 53 9.4 12.4 10.9 200 18.3 SHCC 2 18 4.0 4.0 4.0 50 12.5 SHCC 3 18 4.0 5.6 4.8 50 10.4 SHCC 4 14 2.5 4.2 3.35 31 9.3 SHCC 5 18 5.0 6.6 5.8 50 8.6

Note: All data are assumed values, in the normal range of test results at UM and the JCI round robin tests (Kanakubo 2006). For all cases, tensile and compressive strain capacity are assumed to be 5% and 0.5%.

00.10.20.30.40.50.60.70.80.9

1

0 1 2 3 4 5 6Tensile strain (%)

Nor

mal

ized

dist

ance

from

ext

rem

e . te

nsio

n fib

er to

neu

tral a

xis

SHCC 1SHCC 2SHCC 3SHCC 4SHCC 5

Fig. 8 Normalized distance from extreme tension fiber to neutral axis – tensile strain relation for SHCCs (distance normalized by the beam height).

0.8

0.9

1

4 5 6


this study is shown in Table 5, including PVA-ECC 1, 2 and 3. These SHCC materials feature high amount of fly ash in the mix proportion, with fly ash to cement ratios of 1.2, 2.0, and 2.8, respectively. Additionally, PVA-ECC 4 and Ductal from JCI round robin test (Kanakubo 2006) are also listed in Table 5, which will be used for comparison between UM method and JCI method.

A Hobart mixer was used in this investigation, with a full capacity of 12 liters. All beam, uniaxial tensile and compressive specimens were cast from the same batch. The beam and uniaxial tensile specimens were cast hori-zontally and compressive cylinder specimens were cast vertically. At least 3 specimens were prepared for each test. After demolding, all specimens were cured in a sealed container with about 99% humidity under room temperature for 28 days before testing.

Four point bending test was conducted with a MTS 810 machine. The beam specimen has a dimension of 356mm long, 50 mm high, and 76 mm deep, all dimen-sions are more than 5 times that of the PVA fiber length (8mm), which is the largest length scale among the in-gredients of PVA-ECC. The loading span between two supports is 305mm with a constant moment span length of 102mm. The beam was tested under displacement control at a loading rate of 0.02 mm/second. The flex-ural stress was derived based on simple elastic beam theory and the beam deflection at the loading points was measured from machine displacement directly. The test setup is shown in Fig. 9 (a) in comparison with the JCI method (Fig. 9 (b)).

As shown in Fig. 10, uniaxial tensile test (UTT) was also carried out to directly verify the derived tensile

strain capacity from four point bending test. The coupon specimen used herein measured 304.8 x 76.2 x 12.7 mm. Aluminum plates were glued at both ends of the coupon specimen to facilitate gripping (both ends are fixed). Tests were conducted in an MTS 810 machine with a 25 kN capacity under displacement control, with a loading rate of 0.0025mm/second throughout the test. Two ex-ternal linear variable displacement transducers (LVDTs) were attached to specimen surface with a gage length approximately 180mm to measure the displacement.

4.2 Experimental results The material tensile and compressive properties for dif-

Table 5 Mix proportion for different SHCCs.

Cement Sand Fly ash

Water/cementitious materials Superplasticizer Fiber

PVA-ECC 1 1 0.8 1.2 0.27 0.013 0.02 PVA-ECC 2 1 1.1 2 0.26 0.014 0.02 PVA-ECC 3 1 1.4 2.8 0.26 0.016 0.02

PVA-ECC 4* - - - 0.46 - 0.019 Ductal* - - - 0.22 - 0.02

Note: *: Data from JCI round robin test (Kanakubo, 2006)

(a) (b)

Fig. 9 Comparison of test setup for the (a) UM method and (b) JCI method (Beam deflection at the loading points was obtained from machine displacement directly in UM method).

Bearing

LVDT

Test Specimen

Fig. 10 Setup for uniaxial tensile test.

LVDT LVDT

180

mm


ferent SHCCs can be found in Table 6. With increasing amount of fly ash in PVA-ECC 1-3, the compressive strength continues to decrease as expected, yet PVA-ECC 3 still has a compressive strength of about 38 MPa. For all SHCCs the typical coefficient of variations (COV) of first cracking strength and ultimate tensile

strength are less than 15%, similar to that of compres-sive strength. Conversely, the COV of tensile strain ca-pacity are in the range of 26%-60% except for PVA-ECC 2 and 3, where the robustness of tensile ductility increased (in the form of reduced COV) due to the us-age of high volume fly ash (Wang 2005). This general trend – relatively low COV for tensile strength and high COV for tensile strain capacity can also be found in Kanakubo (2006). This further confirmed the rationale of quality control for the tensile strain capacity instead of tensile strength.

The flexural stress-load point deflection curves for PVA-ECC 1 were shown in Fig. 11. If the specimen was not fully contacted with the test apparatus, the initial loading stage may show unrealistic low stiffness in some case as shown in Fig. 11 (a). This can be easily corrected by discounting this part of deflection from the load point deflection, as revealed in Fig. 11 (b). The initial deflection with low stiffness, if any, is typically less than 0.2 mm, comprising of less than 5% of the deflection capacity.

Similar to PVA-ECC1 in Fig. 11, PVA-ECC 2 and 3 also show typical deflection hardening behavior under FPBT, with the bottom surface of the specimens after bending test shown in Fig. 12. More and more saturated microcrack is revealed from PVA-ECC 1 to 3, along with gradual increase of deflection capacity (Table 7).

The modulus of rupture for PVA-ECC 1-3 ranges from 10-12 MPa, about 2.4-3.0 times that of their first cracking strength. This is consistent with the finding of Maalej and Li (1994) that this ratio should be about 2.7 for elastic-perfectly plastic material (for tensile portion of beam), such as the PVA-ECCs investigated in this study.

Table 6 Material tensile and compressive properties from experiment for different SHCCs.

First cracking

strength (MPa)

Ultimate tensile strength

(MPa)

Tensile strain capacity

(%)


PVA-ECC 1 4.6±0.3 (7%) 5.3±0.6 (11%) 2.1±1.1 (52%) 54.6±6.5 (12%) PVA-ECC 2 3.9±0.5 (13%) 4.6±0.2 (4%) 3.5±0.3 (9%) 46.0±3.8 (8%) PVA-ECC 3 4.0±0.2 (5%) 4.9±0.1 (2%) 3.7±0.4 (11%) 37.5±1.7 (5%)

PVA-ECC 4* 3.7±0.8 (21%) 5.0±0.5 (10%) 2.7±0.7 (26%) 31.3±0.8 (3%) Ductal* 13.7±0.9 (7%) 15.3±1.0 (7%) 0.5±0.3 (60%) 198.0±3.7 (2%)

Note: *: Experimental data from JCI round robin test (Kanakubo, 2006); Number in parenthesis is coefficient of variation (COV).

Table 7 Comparison between predicted tensile strain capacity from FPBT and tensile strain capacity from UTT.

Tensile strain capacity from

UTT (%)

Deflection capacity

from FPBT (mm)

Predicted tensile strain capacity (%)

Difference between pre-diction and test result

(%) PVA-ECC 1 2.1±1.1 5.8±1.6 2.4±0.8 14% PVA-ECC 2 3.5±0.3 7.0±1.3 3.3±0.5 -6% PVA-ECC 3 3.7±0.4 9.3±0.9 4.4±0.1 19%

FA/C=1.2

02468

1012141618

0 2 4 6 8 10 12Load point deflection (mm)

Flex

ural

stre

ss (M

Pa) .

(a) FA/C=1.2

02468

1012141618

0 2 4 6 8 10 12Load point deflection (mm)

Flex

ural

stre

ss (M

Pa) .

(b)

Fig. 11 Experimental flexural stress – load point deflec-tion relation for PVA-ECC 1 (a) before and (b) after cor-rection of the initial loading stage.

PVA-ECC

PVA-ECC


5. Validation and verification of the proposed method

To validate the proposed inverse method, the deflection capacity obtained from FPBT is converted to tensile strain using Equations (4) and (5) (derived for the same beam size as used in the FPBT experiments) and then compared with tensile strain capacity obtained directly from uniaxial tensile test for PVA-ECC 1-3. As revealed in Table 7 and Fig. 13, the tensile strain capacity de-rived from FPBT predicts the uniaxial tensile test results with reasonable accuracy, with less than 20% difference. This agreement demonstrates the validity of the pro-posed inverse method.

To further verify the proposed UM method, compari-son between UM method and JCI method was con-ducted based on JCI round robin test data (Kanakubo 2006). As mentioned previously, bending test results from JCI round robin test are presented in the form of moment –curvature relation. To facilitate the compari-son, the curvature capacity can be converted to tensile strain capacity using Equations (6) and (7) in UM method. Within the JCI method, the tensile strain capac-ity is obtained by solving following equations (JCI-S-003-2005):

)1(, nlubtu xD −⋅⋅= φε (8)

0123 23 =−+ ∗mxx nlnl (9)

3max

DBEM

mu ⋅⋅⋅

=∗

φ (10)

where btu ,ε is the predicted tensile strain capacity (%); uφ is the curvature capacity (1/mm), which can be calculated from two LVDTs measurements (Fig. 9 (b)); D is depth of the test specimen (=100 mm); nlx is the ratio of the distance from compressive edge (extreme

compression fiber) to neutral axis over depth of test specimen, which needs to be solved from Equation (9); Mmax is maximum moment )( mmN ⋅ ; E is the static modulus of elasticity (N/mm2); B is the width of test specimen (100 mm). For more details, readers are re-ferred to the Appendix to JCI-S-003-2005.

As shown in Table 8 and Fig. 14, predictions based on both the UM method and the JCI method reveal comparable results with those from uniaxial tensile tests. Furthermore, the UM method shows slightly smaller discrepancy with the uniaxial tensile test result (Table 8) based on limited data. The consistency between the UM method and the JCI method and verification by independent JCI round robin test data further demon-strate the validity of the proposed UM method.

The advantage of the UM method over the JCI method lies in its simplicity, both in the experiment and data interpretation phases. In the experiment phase, the UM method only requires machine displacement to be measured. This is not the case for the JCI method, where complicated setup such as LVDTs is needed to measure curvature, as revealed in Fig. 9 (a) and (b). In

0

1

2

3

4

5

6

1.2 2 2.8FA/C

Tens

ile s

train

cap

acity

(%) .

Tensile test resultsPrediction

Fig. 13 Comparison of tensile strain capacity from UTT test with prediction from proposed UM method for differ-ent PVA-ECCs.

PVA-ECC 1 PVA-ECC 2 PVA-ECC 3

0

1

2

3

4

5

6

1 2Different SHCCs

Tens

ile st

rain

cap

acity

(%) . Tensile test results

Prediction (JCI method)Prediction (UM method)

Fig. 14 Comparison of tensile strain capacity from UTT with predictions from JCI method and proposed UM method for different SHCCs. (Experimental data for both UTT and FPBT are from JCI round robin test and only two FPBT specimens were reported for Ductal.)

PVA-ECC 4 Ductal

Fig.12 Beam bottom surface shown increasingly satu-rated microcracks after bending test for PVA-ECC 1-3 (Microcracks are magnified with thick pen for the con-stant moment span length only).

(a) PVA-ECC 1

(b) PVA-ECC 2

(c) PVA-ECC 3


the data interpretation phase, the UM method only needs a simple master curve or linear equation to con-vert deflection capacity directly into tensile strain capac-ity, while the JCI method requires relatively compli-cated procedures (solving cubic equation) to obtain ten-sile strain capacity. Considering the large amount of specimens needed to be tested during construction, the UM method seems to be more suitable for quality con-trol purpose due to its simplicity, efficiency and reason-able accuracy.

6. Conclusions

To facilitate the quality control of the strain hardening cementitious composites on site, a simplified inverse method is proposed to covert the deflection capacity from simple beam bending test to tensile strain capacity through linear transformation. The linear transformation (in the form of master curves) is derived from paramet-ric study with a wide range of parametric values of ma-terial tensile and compressive properties based on a theoretical model. This proposed method has been ex-perimentally validated with uniaxial tensile test results with reasonable agreement. In addition, this proposed method compares favorably with the JCI method in ac-curacy, but without the associated complexity.

The following specific conclusions can be drawn from this study: 1. A simple inverse method has been successfully

developed to derive tensile strain capacity of SHCC from beam bending deflection capacity by using a master curve. This method is expected to greatly ease the on-site quality control for SHCC in terms of much simpler experiment setup re-quirement (compared with both UTT and the JCI inverse method) and data interpretation procedure (compared with the JCI method), yet with reason-able accuracy (within 20%);

2. The master curve features simple linear transfor-mation with a narrow band width. The master curve decouples the dependence of tensile strain capacity on the moment capacity in contrast with the JCI method where tensile strain capacity is de-pendent on both curvature capacity and moment capacity. Therefore, this method allows simple lin-ear equations (Equation (4) and (5)) to be used for

easy data interpretation; 3. A linear relation between the deflection capacity

and the tensile strain capacity is observed based on parametric studies. It has been shown that this lin-ear relation is closely related to the fact that the neutral axis of the SHCCs under bending rapidly approaches the extreme compression fiber and quickly stabilizes (the plateau distance from ex-treme tension fiber to neutral axis is about 90% of the beam depth);

4. All linear curves relating tensile strain capacity and deflection capacity lie in a narrow band regardless of actual material properties. This suggests that beam deflection capacity is most sensitive to ten-sile strain capacity for a given FPBT geometric dimensions, and much less sensitive to other prop-erties such as compressive strength, Young’s Modulus, etc. This could be explained by the fact that the distance from extreme tension fiber to neu-tral axis stabilizes to about 90% of the beam depth for all SHCCs with different material properties.

It should be noted that the following assumptions are made when the proposed UM method is used: (a) The tested material is truly a strain hardening type; (b) The major target for quality control for this material is ten-sile ductility; and (c) For this method to be most effec-tive, a standardized beam with fixed geometric dimen-sions should be agreed upon by the user community.

Acknowledgements The authors thank the Michigan Department of Trans-portation, and the National Science Foundation for funding portions of this research (CMS-0223971, CMS-0329416).

References JCI-S-003-2005. (2005). “Method of test for

bendingmoment–curvature curve of fiber reinforced cementitious composites.” Japan Concrete Institute Standard, 7.

Kanakubo, T. (2006). “Tensile characteristics evaluation method for DFRCC.” Journal of Advanced Concrete Technology, 4 (1), 3-17.

Kanda, T., Kanakubo, T., Nagai, S. and Maruta, M. (2006). “Technical consideration in producing ECC Pre-cast structural element.” Proceedings of

Table 8 Comparison between uniaxial tensile test results with predictions based on the JCI method and the UM method.

Tensile strain capacity from

UTT (%)

Curvature Capacity

from FPBT (µ/mm)

Predicted tensile strain capacity

(JCI method) (%)

Predicted tensile strain capacity

(UM method) (%)

PVA-ECC 4 2.7±0.7 349.2±96.3 3.1±0.9 (15) 3.0±1.1 (11) Ductal 0.5±0.3 85.0* 0.6* (20) 0.5* (0)

Note: *: Only two bending specimens were reported. The number in parenthesis is the difference (in percentage) between the predictions and test results from uniaxial tensile test.


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Ostergaard, L., Walter, R. and Olesen, J. F. (2005). “Method for determination of tensile properties of engineered cementitious composites (ECC).” Proceedings of ConMat'05, Vancouver, Canada.

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